Determination of Moment Capacity M p for Rectangular Reinforced Concrete Columns 1

Digest 2009, December 2009 1293-1312

Determination of Moment Capacity Mp for Rectangular Reinforced Concrete Columns†1 Cem AYDEMİR* Mustafa ZORBOZAN** Sema NOYAN ALACALI*** ABSTRACT

Turkish Seismic Code stipulates capacity design procedure for the calculation of design shear force of RC beams and columns. Therefore, the plastic hinge moment capacity (Mp) at the ends of columns must be determined as accurately as possible. Turkish Seismic Design Code recommends an increase of 40% in ultimate moment capacity (Mr) to obtain the plastic hinge moment capacity (Mp), unless a detailed calculation is carried out. However, this recommendation remains insufficient in attaining plastic hinge moment capacity for high levels of axial load.

In this study, new analytical equations have been derived to obtain the plastic hinge moment capacity of rectangular RC columns in an accurate and practical way. Plastic hinge moment capacity of columns calculated with the proposed equations is compared to the capacity obtained from experimental results. It is shown that the proposed equations yield accurate results.

Keywords: Column moment capacity, M-N interaction diagrams, strain hardening, confined concrete

1. INTRODUCTION

In line with the requirement for capacity design procedure in the Turkish Seismic Design Code, the design shear force of R/C columns should be calculated based on the flexural moment capacities at the ends of columns (Figure 1). Therefore, the plastic hinge moment capacity (Mp) at the end of the column must be determined as accurately as possible.

The moment capacity of a R/C member under pure bending or combined bending and axial force can be calculated taking into consideration different assumptions. Three different definitions of moment capacities are summarized below.

Mr , Ultimate Moment Capacity (TS500) [1]

Ultimate moment capacity is obtained using design strengths of materials ( cckcd /ff γ= , sykyd /ff γ= ) with the assumptions given by TS500 [1]. The assumptions for

* Yildiz Technical University, İstanbul, Turkey, [email protected] ** Yildiz Technical University, İstanbul, Turkey, [email protected] *** Yildiz Technical University, İstanbul, Turkey, [email protected] † Published in Teknik Dergi Vol. 20, No. 1 January 2009, pp: 4545-4565

Determination of Moment Capacity Mp for Rectangular Reinforced Concrete Columns

1294

calculation of ultimate strength can be summarized as: Plane sections remain plane after bending, tensile strength of concrete is neglected, steel is an elasto-plastic material, stress distirubition at the compresion zone is expressed as an equivalent rectangular stress block and maximum strain in the extreme fiber of concrete in compression is 0.003.

Mn , Nominal Moment Capacity (ACI-318) [2]

Nominal moment capacity given by ACI-318 [2] is obtained with characteristic material strengths. In design process nominal moment capacity is reduced with a strength reduction factor (φ).

Mp , Flexural Moment Capacity

For a flexure dominant structural element, flexural moment capacity calculated based on code assumptions (Mr, Mn), is always less than the actual capacity (Mp) since strain hardening of reinforcing steel is neglected and confinement effect of concrete is overlooked and material strengths are reduced with partial safety factors.

The maximum flexural moment capacity (Mp) can be defined as the maximum moment obtained from moment-curvature analyses considering strain hardening of steel, crushing of cover concrete, tensile strength of concrete and stress-strain relationship of confined concrete. However; calculation of Mp through moment-curvature analysis during the design process is not always practical and useful.

Figure 1: Bending moment values of columns for based on capacity design procedure in the

Turkish Seismic Design Code [3, 4]

Moment curvature relations of a sample column section, under various axial load levels, are shown in Fig. 2. In Fig. 3, Mp moment capacities of sample column sections with corresponding axial load levels are given as an interaction diagram (PEMKED) [5].

Modified Kent-Park model for confined and unconfined concrete under compression, a strain hardening material model for reinforcement steel, a material model considering tensile strength of concrete are used to obtain moment curvature relations as well as a fiber model that is used for geometrical definition of the section [5]. Maximum stress, strain corresponding to maximum stress and effective maximum strain for unconfined concrete are assumed to be 0.85.fck, 0.002 and 0.004, respectively [6].

nl

Mpü

Mpa

( ) npapüe /MMV l+=

Ve Ve

Cem AYDEMİR, Mustafa ZORBOZAN, Sema NOYAN ALACALI

1295

Figure 2: Moment curvature relations for various axial load levels

Figure 3: Mp moment capacities with corresponding axial load levels [5]

In Turkish Seismic Design Code, it is designated that maximum bending moment capacity Mp can be taken as 1.4Mr [3, 4], unless detailed calculations are performed. Although, the value of Mp/Mr ratio is given as 1.4, regardless of column axial load level, this ratio is influenced by many design variables primarily the column axial load level [5, 6]. Due to this fact, Mp/Mr≈1.4 assumption is sometimes unrealistic with increasing axial load [5, 6].

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3

N = 0 kN (A)

N = 625 kN (B)

N = 1250 kN (C)

N = 4000 kN (E)

N = 2500 kN (D)

Mom

ent (

kNm

)

Curvature (rad/m)

Column: S1

Ac= 500 x 500 mm2 fck= 20 MPa, fctk=1.6 MPa fyk = fywk = 420 MPa As = 2514 mm2(8φ20) Stirrups: 2-φ8/100 mm εsh=0.01, εsu=0.10, fsu=500 MPa

0

1000

2000

3000

4000

5000

6000

7000

0 100 200 300 400 500 Moment (kNm)

AB

C

D

E

PEMKED

Axi

al lo

ad (k

N)

Column: S1


1296

Ersoy has carried out analytic studies on obtaining column moment capacities Mp, with a more realistic approach than 1.4Mr approach of TSDC [6]. Ersoy showed that confinement effect in concrete is more effective than strain hardening of reinforcement steel on column moment capacities Mp, and recommended use of fyk and fcc (fcc=1.15fck) instead of fyd and fcd. This approach yields more accurate results for increasing levels of axial load, however; it underestimates the moment capacity for lower levels of axial load since the effect of strain hardening on moment capacity can not be considered in this approach [5].

In the current study, more realistic and practical approaches to obtain column moment capacities Mp, are proposed along with comparisons with experimental results.

2. EXPERIMENTAL COMPARISONS

In this section; test results of columns subjected to constant axial load and cyclic flexure simulating earthquake loading, found in the literature, have been considered. Mp moment capacities obtained from test elements (Mp,experimental=VmaxL), are compared with the moment capacities calculated based on the approximate methods mentioned above.

Schematical views of test mechanisms, typical column sections and general properties of test elements used in comparisons are summarized in Fig. 4, Table 1 and Table 2, respectively.

Figure 4: Schematical view of test mechanisms

Table 1: Typical column sections of test columns

Typical column section

1 2 3 4 5 6 7

N

VL

N

2V

L

L

N

V L


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Table 2: General properties of test columns

No Reference/ Element

Type b/h (mm/mm)

ρt fck / fyk/ fsu (MPa)

fywk (MPa)

Stirrups Ø/s(mm)

n

1 [7]/No1 1 550/550 0.0179 23.1/375/635.6 297 φ10/80 0.26

2 [7]/No2 1 550/550 0.0179 41.4/375/635.6 316 φ12/75 0.214

3 [7]/No3 2 550/550 0.0179 21.4/375/635.6 297 φ10/75 0.42

4 [7]/No4 2 550/550 0.0179 23.5/375/635.6 294 φ12/62 0.60

5 [8]/No3 2 400/400 0.0151 23.6/427/670 320 φ12/100 0.38

6 [8]/No4 2 400/400 0.0151 25/427/670 280 φ10/90 0.21

7 [9]/No1 1 400/400 0.0151 46.5/446/702 364 φ7/85 0.10

8 [9]/No2 1 400/400 0.0151 44/446/702 360 φ8/78 0.30

9 [9]/No3 1 400/400 0.0151 44/446/702 364 φ7/91 0.30

10 [9]/No4 1 400/400 0.0151 40/446/702 255 φ6/95 0.30

11 [10]/No9 3 400/600 0.0188 26.9/432/* 305 φ12/80 0.10

12 [11]/No2 4 400/400 0.0157 25.6/474/721 333 φ12/80 0.20

13 [11]/No4 4 400/400 0.0157 25.6/474/721 333 φ12/80 0.20

14 [11]/No5 2 550/550 0.0125 32/511/675 325 φ12/110 0.10

15 [11]/No6 2 550/550 0.0125 32/511/675 325 φ12/110 0.10

16 [12]/85STC-1 5 250/250 0.0162 27.9/374/494 506 φ5.5/50 0.106

17 [12]/85STC-2 5 250/250 0.0162 27.9/374/494 506 φ5.5/50 0.106

18 [12]/85STC-3 5 250/250 0.0162 27.9/374/494 506 φ5.5/50 0.106

19 [12]/85PDC-1 5 250/250 0.0162 24.8/374/494 352 φ5.5/50 0.106


1298

Table 2: General properties of test columns (continued)

No Reference/

Element

Type b/h (mm/mm)


fywk (MPa)

Stirrups Ø/s(mm)

n

20 [12]/85PDC-2 5 250/250 0.0162 27.9/374/494 506 φ5.5/50 0.106

21 [12]/85PDC-3 5 250/250 0.0162 27.9/374/494 506 φ5.5/50 0.106

22 [13]/No. 1S1 6 305/305 0.0163 29.1/367/578 363 φ9.5/76 0.099

23 [13]/No. 2S1 6 305/305 0.0163 30.7/367/578 363 φ9.5/127 0.093

24 [13]/No. 5S1 6 305/305 0.0163 29.4/429/657 392 φ9.5/76 0.195

25 [13]/No. 6S1 6 305/305 0.0163 31.8/429/657 392 φ9.5/127 0.181

26 [14]/U-1 5 350/350 0.0321 43.6/430/* 470 φ10/150 0.0

27 [14]/U-4 5 350/350 0.0321 32/438/* 470 φ10/50 0.153

28 [14]/U-6 4 350/350 0.0321 37.3/437/* 425 φ6.4/65 0.131

29 [14]/U-7 4 350/350 0.0321 39/437/* 425 φ6.4/65 0.126

30 [15]/C1-1 2 400/400 0.0214 24.9/497/592 459.5 φ6.3/50 0.113

31 [15]/C1-2 2 400/400 0.0214 26.7/497/592 459.5 φ6.3/50 0.158

32 [15]/C1-3 2 400/400 0.0214 26.1/497/592 459.5 φ6.3/50 0.216

33 [15]/C2-1 2 400/400 0.0214 25.3/497/592 459.5 φ6.3/52 0.111

34 [15]/C2-2 2 400/400 0.0214 27.1/497/592 459.5 φ6.3/52 0.156

35 [15]/C2-3 2 400/400 0.0214 26.8/497/592 459.5 φ6.3/52 0.21

36 [15]/C3-1 2 400/400 0.0214 26.4/497/592 459.5 φ6.3/54 0.107

37 [15]/C3-2 2 400/400 0.0214 27.5/497/592 459.5 φ6.3/54 0.153 * Ultimate strength of steel is not given.


1299

Table 2: General properties of test columns (continued)

No Reference/

Element

Type b/h (mm/mm)


fywk (MPa)

Stirrups Ø/s(mm)

n

38 [15]/C3-3 2 400/400 0.0214 26.9/497/592 459.5 φ6.3/54 0.209

39 [16]/L1 7 400/400 0.0142 24.8/362/* 325 φ9/100 0.032

40 [16]/L2 7 400/400 0.0142 24.8/362/* 325 φ9/100 0.032

41 [17]/No7 2 400/400 0.0151 28.3/440/674 466 φ10/117 0.223

42 [17]/No8 2 400/400 0.0151 40.1/440/674 466 φ10/92 0.39

43 [18]/No-5 1 400/400 0.0151 41/474/633.3 372 φ8/81 0.50

44 [18]/No-6 1 400/400 0.0151 40/474/633.3 388 φ7/96 0.50

45 [19]/BG-1 4 350/350 0.0195 34/455.6/660 570 φ9.5/152 0.428

46 [19]/BG-2 4 350/350 0.0195 34/455.6/660 570 φ9.5/76 0.428

47 [19]/BG-3 4 350/350 0.0195 34/455.6/660 570 φ9.5/76 0.20

48 [19]/BG-4 2 350/350 0.0293 34/455.6/660 570 φ9.5/152 0.462

49 [20]/c5-40N 6 203/203 0.0193 38.1/572/729.1 513.7 φ9.5/76 0.362

50 [21]/D1N60 2 250/250 0.0243 37.6/461/634.3 485 φ4/40 0.60 * Ultimate strength of steel is not given.

The ratios of the moment capacity (Mp,experimental=VmaxL), obtained from maximum base shear occuring at the ends of test columns with properties given in Table 2 to moment capacity obtained with 1.4Mr assumption given by TSDC and their variations with column axial load level are shown in Fig. 5.

It can be seen from Fig. 5 that, the ratios of Mp,experimental/1.4Mr – in other words – the ratios of maximum shear forces occuring at the ends of elements to shear forces obtained with 1.4Mr assumption (Vmax/Ve), vary between 0.81~2.1 and Mp,experimental/1.4Mr ratio increases for increasing dimensionless axial load level. It is a normal tendency because of ultimate moment (Mr) becomes less than the actual capacity and decreases from a lower axial load level (balanced axial load), for increasing dimensionless axial load levels. The increase in Mp,experimental/1.4Mr ratios can be much more than that given above for axial load levels higher than the maximum value considered above. In Fig. 6, Mp,experimental moment


1300

capacity (actual capacity) is marked for axial load level of ≈0.7, on the interaction diagram obtained with 1.4Mr assumption for a column section to explain this behaviour briefly. The ratio of Mp,experimental/1.4Mr is nearly 66 for axial load level n≈0.7. This result is very interesting even it is unusual in practice.

Figure 5: Variation of Mp,experimental/1.4Mr ratio with column axial load level

Figure 6: Mp,experimental/1.4Mr ratio for higher axial loads

0.6 0.8

1.0 1.2

1.4 1.6

1.8 2.0

2.2

0.00 0.10 0.20 0.30 0.40 0.50 0.60 n=Nd/(bhfck)

Mp,

exp

erim

enta

l /1.

4Mr

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.00 0.05 0.10 0.15 0.20 0.25 0.30

TSDC′07 (1.4Mr)

Actual capacity [18],Mp=468.8kNm N=4704 kN, Vmax=293 kN, L=1.6m

Watson ve Park 1989/No-7

Ac= 400 x 400 mm2 fck= 42 MPa fyk = 474 MPa ρt = 0.0151

M/(bh2fck)

n=N

/(bhf

ck)


1301

The ratio of Mp,experimental/1.4Mr vary between 0.81~1.1 for axial load levels lower than 0.3 as can be seen in Fig.5. Mean ratio is 0.97 and the error range is about ±10% for these axial load levels.

The comparisons of moment capacities obtained with the method proposed by Ersoy (fcc=1.15fck) with the experimental moment capacities (Mp,experimental=VmaxL) are shown in Fig. 7.

Figure 7: Variation of Mp,experimental /MErsoy ratio with column axial load level

It can be seen from Fig. 7 that Mp,experimental/ MErsoy ratios vary between 0.93~1.26 and the error rate increases for decreasing axial load level. The authors find it beneficial to investigate the reasons of this increase. Ersoy has emphasized that, confined concrete strength can not be less than 1.15fck and moment capacities calculated with this assumption are less than the real capacities for columns having minimum confinement given in TSDC. Besides, Ersoy has highlighted that, moment capacities calculated for confined concrete strength equal to 1.15fck , can be considerably less than the real capacity for columns having confinement more than the minimum value [6]. As a matter of fact, there are some columns having confined concrete strength greater than 1.15fck in Fig. 7. The effect of increase in concrete strength on column moment capacity is limited for relatively lower axial load levels. The test elements of 26, 27 and 28 (U-1, U-4, U-6) in Table 2 can be given as an example. Concrete strengths of these elements should be increased 6.2, 2.5 and 3.6 times, respectively, to reach experimental moment capacities with the TS-500 approach mentioned above. These increased values are not realistic for confined concrete strengths. The increase in concrete strength begin to show a decrease tendency and get close to confined concrete strength for increasing axial load levels. In this context, not taking into consideration the stress increase because of strain hardening effect will lead to underestimation of moment capacities for low axial load levels. The moment capacities

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.00 0.10 0.20 0.30 0.40 0.50 0.60 n=Nd/(bhfck)

Mp,

exp

erim

enta

l /M

p, Ers

oy

TS-500 approach fcd=1.15fck, fyd=fyk [6]


1302

obtained by Ersoy’s method are considerably close to real moment capacities for axial load levels higher than 0.25 and error rates are less than 10%.

3. PROPOSED METHODS TO OBTAIN COLUMN MOMENT CAPACITIES, Mp

In this section, two simple procedures to obtain column moment capacities are proposed. In the first method, column moment capacity is obtained based on the ultimate moment, whereas increased concrete and steel strengths are used in the second method.

The investigations made in the previous section show that, 1.4Mr approach and Ersoy’s method provide good results for specific axial load levels (Fig. 5 and 7). An equation can be derived to obtain column moment capacity due to axial load level based on the relationhip between ultimate moment capacity and moment capacities found by Ersoy’s method.

With this purpose, equations proposed by Çakıroğlu and Özer as functions of interaction curves for symmetrically reinforced rectangular R/C columns based on ultimate design will be used as below [22].

• For material design strengths (fcd=fck/1.5 ; fyd=fyk/1.15) ;

ck

d

fhbNn

⋅⋅= ,

ck2

d

fhbMm

⋅⋅= (1)

st ykt

1 2 ck

A fω

k k b h f⋅

=⋅ ⋅ ⋅ ⋅

(2)

for n < 0.2;

n48.1n92.2m86.2 2t ⋅−⋅+⋅=ω (3a)

for 0.2 < n ≤ 0.3;

tω 2.92 m 0.192= ⋅ − (3b)

for n > 0.3;

2tω 2.92 m 0.62 n 0.254 n 0.32= ⋅ + ⋅ + ⋅ − (3c)

• For any given value of material coefficents;

c

d

fhb85.0Nn

⋅⋅⋅= ,

c2

d

fhb85.0Mm

⋅⋅⋅= (4)


1303

c21

ystt fhb85.0kk

fA⋅⋅⋅⋅⋅

⋅=ω (5)

for n < 0.35;

n29.1n44.1m49.2 2t ⋅−⋅+⋅=ω (6a)

for 0.35 ≤ n ≤ 0.50;

295.0m54.2t −⋅=ω (6b)

for n > 0.50;

49.0n22.0n305.0m54.2 2t −⋅+⋅+⋅=ω (6c)

Figure 8: k1 coefficients for different longitudinal reinforcement configurations[22]

zs/h=0.90

zs/h=0.85

zs/h=0.80

zs/h=0.75

zs/h=0.70

0.85

0.90

0.95

1.00

1.05

1.10

1.15

0 0.5 1 1.5 2 2.5 3

m/n

k2

Figure 9: k2 coefficients for different zs/h ratios [22]

Type 1

λ=0.50

Type 2 λ=0.25

Type 3 λ=0.125

Type 4

λ=0.0 1.00

1.10

1.20

1.30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Type 1

Type 2

Type 3

Type 4

n=Nd/(bhfck)

k1


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In (1∼6) equations; Ast is the total longitudinal reinforcement area, ωt is the mechanical reinforcement index, k1 is the coefficient to represent different longitudinal reinforcement configurations, (Fig. 8), k2 is a coefficient to represent different zs/h ratios (Fig. 9), fc is concrete compression strength in case of any given material coefficient, fy is steel strength in any given material coefficient.

The steps to obtain a uniqe Mp,Ersoy/Mr valid for all R/C rectangular columns are summarized below.

Material design strengths (fcd and fyd) are used to obtain ultimate moment capacity (Mr) using Eq.(1-3) for any axial load level, whereas fc=1.15fck and fy=fyk [6] are used to obtain the moment capacity (Mp) in Ersoy’s method using Eq.(4-6).

Equation (7) is derived from Equations (2) and (5) by rearranging two equations according to total longitudinal reinforcement area (Ast), and equalizing to each other and simplifying, respectively. (In equation (5) fc=1.15fck, fy=fyk).

15.185.0tt ⋅⋅ω=ω (7)

For the terms tω and tω in equation (7), functions defined for corresponding axial load

levels (equations (3a, b, c) and (6a, b, c)) are used and n023.1n ⋅= transformation is adopted, so Mp/Mr ratio can be obtained with the equations given below.

2p ckM m 0.85 b h 1.15 f= ⋅ ⋅ ⋅ ⋅ ⋅ (8)

ck

d

fhbNn

⋅⋅= ,

ck2

r

fhbMm

⋅⋅= (9)

mMM 2

1r

p β+β= (10)

In Eq.(10); β1 ve β2 are coefficients depending on axial load level, Mr, that is column ultimate moment capacity, m is a dimensionless column ultimate moment capacity, respectively. Variations of β1 and β2 coefficients for different levels of axial load are given in Table 3.

The Mp,Ersoy/Mr ratios, varying against dimensionless axial load level and dimensionless column ultimate moment capacity are calculated from Eq.(10) for all symetrically reinforced R/C rectangular column sections for an axial load level range of 0.15bhfck ∼ 0.5bhfck regardless of column longitudinal reinforcement configuration and zs/h ratio can be seen in Fig. 10.


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Table 3: Variations of β1 and β2 coefficients against axial load levels

Axial load β1 β2

n<0.2 1.15 0.58n2 - 0.076n

0.2 ≤ n ≤ 0.3 1.173 0.52n - 0.59n2 - 0.08

0.3 < n < 0.34 1.173 0.62n - 0.34n2 - 0.13

0.34 ≤ n ≤ 0.5 1.16 0.10n + 0.24n2 - 0.013

Figure 10: Distributions of p,Ersoy rM / M ratios obtained with Eq.(10) for rectangular R/C

columns

As it can be seen in Fig. 5, 1.4Mr approach provides good results for low axial load levels. The moment capacities obtained with this approach are less than real moment capacities of columns for increasing axial load levels. Moment capacities obtained with the method proposed by Ersoy provide very close results to actual capacity for these axial load level ranges. As a result, it is meaningful to use 1.4Mr approach for axial load levels lower than 0.25, whereas Ersoy’s method can be used for axial load levels higher or equal to 0.25.

The coefficients used in Eq.(10) vary according to the axial load level. Besides, these coefficients can be defined as a single equation for axial load level higher or equal to 0.25 in a similar form of Eq.(10). For this purpose, Eq.(11) is derived by Statistica program [23].

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

m=Mr/(bh2fck)

n=Nd/(bhf

ck)

Mp,Ersoy/Mr =

m=Mr/(bh2fck)

n=N

d/(bh

f ck)


1306

m07.0n34.016.1

MM

r

p −⋅+= (11)

The comparisons for Mp,Ersoy/Mr ratios obtained from Eq.(11) with Eq.(10) are shown in Fig. 11 for axial load levels higher or equal to 0.25. The coefficient of correlation of Eq.(11) is 0.998.

Figure 11: Comparisons of Mp,Ersoy/Mr ratios obtained with Eq. (10) and Eq.( 11) for n≥0.25

From this point of view, column moment capacities Mp can be obtained with equation (12). Values of β coefficient in Eq.(12) for corresponding axial load level are given in Table 4.

m4.1

MM

r

p β+= (12)

Table 4: Variations of β coefficient against axial load level

Axial load β

n<0.25 0

0.25 ≤ n ≤ 0.5 (0.34n - 0.24m - 0.07) ≥ 0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

Mp,Ersoy/Mr ((11) bağıntısıyla)

1.01.21.41.61.82.02.22.42.62.83.03.23.4

Mp,

Ers

oy/M

r ((1

0) b

ağın

tısıy

la)

(Eq.11)

(Eq.

10)


1307

Comparisons for experimental moment capacities (Mp,experimental=VmaxL) with proposed moment capacities by Eq.(12) of test columns with given properties in Table 2 are shown in Fig. 12.

Figure 12: Comparisons for experimental moment capacities with proposed moment

capacities by Eq.(12)

In the second proposed method to obtain column moment capacity, a similar approach to Ersoy’s method is used. The effect of strain hardening on increase in moment capacities should be taken into consideration for decreasing axial load level as can be seen in Fig.7. In this context, strain hardening effect is thought to be taken into consideration by increasing yield strength and experimental moment capacities are used with this purpose. Confined concrete strength can be taken as 1.15fck as proposed by Ersoy’s method.

Figure 13: Variations of coefficents k against axial load levels

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

0.00 0.10 0.20 0.30 0.40 0.50

Mp,

expe

rim

enta

l/Mp,

prop

osed

n=Nd/(bhfck)

Mr approach

(Eq.12)

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

0.00 0.10 0.20 0.30 0.40 0.50

k

n=Nd/(bhfck)

k 1.19=


1308

In Fig.13, variations of coefficent k which represents the increase in yield strength of steel against the axial load level can be seen. The mean value of the ratio of ultimate strength to yield strength (fsu/fyk) is 1.42 for test columns given in Table 2. In this proposed method, the increase in the amount of yield strength (k) is equal to 1.19 which is the mean value of results shown in Fig.13. It is well known that the effect of strain hardening on moment capacity shows a decreasing tendency for increasing axial load level. The similar tendency can be observed for coefficents k in Fig.13.

The comparison of the experimental moment capacities with moment capacities calculated by using increased material strenghts (fcd=1.15fck and fyd=1.19fyk) for columns given in Table 2 are shown in Fig.14 with an error band of 10%. It can be seen in Fig. 14 that the results of the mentioned approach provide similar results to experimental capacities for all axial load levels and error is usually less than 10%.

Figure 14: The comparisons of the experimental moment capacities with proposed moment

capacities (fcd=1.15fck and fyd=1.19fyk)

4. CONCLUSIONS

In this study, two methods are proposed to calculate Mp moment capacity of R/C rectangular columns for various axial load levels in an accurate and practical way. In the first method, column moment capacity is calculated as a function of the ultimate moment and axial load level. In the second method, column moment capacity is calculated using increased material design strengths with the assumptions given in TS-500.

Results based on comparison of existing and proposed methods with experimental results can be summarized as below.

0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Mp,

expe

rim

enta

l / M

p,pr

opos

ed

n=Nd/(bhfck)

TS-500 approach fcd=1.15fck, fyd=1.19fyk


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• Mp,experimental/Mr ratio varies between 1.13 ∼ 2.93 for axial load level range of 0 ∼ 0.6bhfck. For higher axial load levels, these ratios increase considerably. Thus, shear forces at column ends obtained with 1.4Mr assumption are considerably lower than the shear strength required for ductile behaviour especially for high axial load levels.

• Mp,experimental/Mp,Ersoy ratio varies between 0.93 ∼ 1.26 for axial load levels given above. Especially, axial load levels lower than 0.25, error rates increase and yield lower results than actual capacity in the range of 26%.

• Column moment capacity Mp, can be expressed as a function of ultimate moment Mr and axial load level with the help of analytical expressions of M-N interaction diagrams depending on increased design strengths. Moment capacities obtained with Eq.(12) derived based on the approach mentioned above, provide similar results to actual moment capacities.

• The effect of strain hardening on increase of moment capacities should be taken into consideration in columns for decreasing axial load levels. The effect of strain hardening on moment capacity shows a decreasing tendency for increasing axial load levels.

• Column moment capacity Mp can be calculated using increased strengths (fcd=1.15fck ve fyd =1.19fyk) instead of material design strengths given in TS500. It is also found that moment capacities obtained with this approach provide similar results to actual moment capacities for all axial load levels.

• Characteristic material strengths considered for concrete and steel represent the strength with a probability range falling under of 10%. In other words, in-place material strength will be higher than the characteristic strength with a probability of 90%. From this point of view, statistical investigations should be carried out to take the influence of physical and modelling uncertainities on moment capacity of R/C columns into consideration.

Symbols

b : Width of column section

fcc : Confined concrete compressive strength

fcd : Design compressive strength of concrete

fck : Characteristic compressive cylinder strength of concrete

fctk : Characteristic tensile strength of concrete

fyd : Design yield strength of longitudinal reinforcement

fyk : Characteristic yield strength of longitudinal reinforcement

fywk : Characteristic yield strength of transverse reinforcement

fsu : Ultimate strenght of steel

h : Height of column section

k : Coefficent k which represents the increase in the amount of yield strength of steel


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k1 : Coefficient to represent different longitudinal reinforcement configurations

k2 : Coefficients for different zs/h ratios

L : Shear length

nl : Clear height of column

m : Dimensionless ultimate moment capacity

Mn : Nominal moment capacity

Mp : Moment capacity

Mpa : Ultimate moment capacity calculated at the bottom of column clear height

Mpü : Ultimate moment capacity calculated at the top of column clear height

Mp,Ersoy : Moment capacity based on factored material strength

Mr : Ultimate moment capacity

n : Dimensionless axial load level

N : Axial load

Nd : Factored axial force calculated under simultaneous action of vertical loads and seismic loads

s : Spacing of transverse reinforcement

Ve : Shear force taken into account for the calculation of transverse reinforcement of column

Vmaks : Maximum base shear

εcu : Ultimate compressive strain of unconfined concrete

εsh : The strain at which strain-hardening of steel begins

εsu : The ultimate strain of steel

φ : Curvature

Ø : Bar diameter

γc : Partial safety factors for concrete

γs : Partial safety factors for reinforcement steel

λ : The ratio between the reinforcing steel area located away from either tension or compression side of the section and the total reinforcement area of the column

ρt : Longitudinal column reinforcement ratio

ωt : Mechanical reinforcement index


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References

[1] Requirements for Design and Construction of Reinforced Concrete Structures, TS-500, Turkish Standards Institution, Ankara, 2000

[2] Building Codes Requirements for Reinforced Concrete, ACI-318-02, American Concrete Institute, Detroit, 2002.

[3] Turkish Seismic Design Code, Ministry of Public Works and Settlement, Ankara, 1997.

[4] Specification for Structures to be Built in Disaster Areas, Ministry of Public Works and Settlement, Ankara, 2007.

[5] Aydemir C., Determination of Moment Capacity Mp for Rectangular Reinforced Concrete Columns, MSc Thesis, Yıldız Technical University, Istanbul, 2004

[6] Ersoy U., Moment Capacity of Reinforced Concrete Beams and Columns, Technical Journal of Turkish Chamber of Civil Engineers 1998; 9(4):1781–97

[7] Gill, Wayne Douglas., Park, R., and Priestley, M.J.N., Ductility of Rectangular Reinforced Concrete Columns With Axial Load, Report 79-1, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand,February 1979.

[8] Ang Beng G., Priestley, M.J.N., and Park, R., Ductility of Reinforced Bridge Piers Under Seismic Loading, Report 81-3, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, February 1981.

[9] Soesianawati, M.T., Park, R., and Priestley, M.J.N., Limited Ductility Design of Reinforced Concrete Columns, Report 86-10, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, March 1986.

[10] Park, R., and Paulay, T., Use of Interlocking Spirals for Transverse Reinforcement in Bridge Columns. Strength and Ductility of Concrete Substructures of Bridges, RRU (Road Research Unit) Bulletin 84, Vol. 1, 1990, pp. 77-92.

[11] Tanaka H., and Park R., Effect of lateral confining reinforcement on the ductile behavior of reinforced concrete columns. Report 90-2, Department of Civil Engineering, University of Canterbury, June 1990, 458 pages.

[12] Kanda M, Shirai N, Adachi H, and Sato T., Analytical study on elasto-plastic hysteretic behaviors of reinforced concrete members. Transactions of the Japan Concrete Institute, Vol.10, 1988, pp. 257-264.

[13] Atalay MB, and Penzien J., The seismic behavior of critical regions of reinforced concrete components as influenced by moment, shear and axial force. Report No. EERC 75-19, University of California, Berkeley, December 1975, 226 pages.

[14] Saatcioglu M., and Özcebe, G., Response of reinforced concrete columns to simulated seismic loading. American Concrete Institute, ACI Structural Journal, January - February, 1989, pp. 3-12.


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[15] Mo, Y.L., and Wang, S.J., Seismic Behavior of RC Columns with Various Tie Configurations. Journal of Structural Engineering, ASCE, Vol. 126 No.10, October, 2000, pp. 1122-1130.

[16] Ohno, Tomonori, and Nishioka, Takashi, An Experimental Study on Energy Absorption Capacity of Columns in Reinforced Concrete Structures. Proceedings of the JSCE, Structural Engineering/Earthquake Engineering, Vol. 1, No 2., October 1984, pp. 137-147.

[17] Zahn FA, Park R, and Priestley MJN. Design of reinforced bridge columns for strength and ductility. Report 86-7, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, March 1986, 330 pages.

[18] Watson, S., and Park, R., Design of Reinforced Concrete Frames of Limited Ductility, Report 89-4, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, January 1989, 232 pages.

[19] Saatcioglu, M., and Grira, M., Confinement of Reinforced Concrete Columns with Welded Reinforcement Grids, American Concrete Institute, ACI Structural Journal, Vol. 96, No. 1, January-February 1999, pp. 29-39.

[20] Matamoros, A.B., Study of Drift Limits for High-Strength Concrete Columns, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, Oct 1999.

[21] Kono, Susumu, and Watanabe, Fumio, Damage Evaluation of Reinforced Concrete Columns Under Multiaxial Cyclic Loadings, The Second U.S.-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinfoced Concrete Building Structures, Sap, 2002.

[22] Çakıroğlu, A., and Özer, E., Ultimate Load Design for Reinforced Concrete Rectangular and Circular Sections and Their Order of Accuracy, Technical Journal of Turkish Chamber of Civil Engineers, 1990; 1(1): 25-28.

[23] StatSoft Inc. STATISTICA V.6.0 for Windows. Tulsa, OK, USA;1995.

Determination of Moment Capacity M p for Rectangular Reinforced Concrete Columns 1

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